Extending Polynomial Calculus to $k$-DNF Resolution

We consider two well-known algebraic proof systems: Polynomial Calculus (PC) and Polynomial Calculus with Resolution (PCR), a system introduced in [2] which combines together PC and Resolution. Moreover we introduce an algebraic proof system PCRk , which combines together Polynomial Calculus (PC) and k-DNF Resolution (RESk). This is a natural generalization to RESk of PCR. In the paper we study the complexity of proofs in such systems. First we prove that a set of polynomials encoding a Graph Ordering Principle (GOP(G)) requires PCR refutations of degreeΩ(n). This is the first linear degree lower bound for PCR refutations for ordering principles. This result immediately implies that the size-degree tradeoff for PCR Refutations of Alekhnovich et al. [3] is optimal, since there are polynomial size PCR refutations for GOP(G). We then study the complexity of proofs in PCRk , extending to these systems the lower bounds known for RESk: • we prove that random 3-CNF formulas with a linear number of clauses are hard to prove in PCRk (over a field with characteristic different from 2) as long as k is in o( √ logn/ log logn). This is the strongest daglike system where 3-CNF formulas are hard to prove. • Moreover we prove a strict hierarchy result showing that PCRk+1 is exponentially stronger than PCRk .